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Trigonometry Unit 1 Key Concepts and Identities

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  • Convert decimal degrees to degrees, minutes, and seconds

    Multiply the decimal part by 60 to get minutes; multiply the decimal part of minutes by 60 to get seconds.

  • Convert degrees, minutes, and seconds to decimal degrees

    Decimal degrees = degrees + (minutes ÷ 60) + (seconds ÷ 3600).

  • Complement of an angle

    The angle that, when added to the given angle, equals \(90^\circ\).

  • Supplement of an angle

    The angle that, when added to the given angle, equals \(180^\circ\).

  • Coterminal angles

    Angles that share the same terminal side, differing by multiples of \(360^\circ\) or \(2\pi\) radians.

  • Find angle measures using angle relationships

    Use complementary, supplementary, and coterminal angle rules to determine unknown angles.

  • Find trigonometric function values given a point on the terminal side

    Use coordinates \((x,y)\) and radius \(r=\sqrt{x^2+y^2}\) to find sine, cosine, and tangent.

  • Find trigonometric function values given an equation on the terminal side

    Use the equation to find coordinates or ratios, then calculate trig functions accordingly.

  • Reciprocal identities

    csc = 1/sin, sec = 1/cos, cot = 1/tan.

  • Pythagorean identity

    \(\sin^2\theta + \cos^2\theta = 1\)

  • Quotient identities

    \(\tan\theta = \frac{\sin\theta}{\cos\theta}\) and \(\cot\theta = \frac{\cos\theta}{\sin\theta}\)

  • Find the quadrant of an angle given trig function values

    Use the sign of the trig function to determine the quadrant based on ASTC (All Students Take Calculus) rule.

  • Definition of sine in terms of a point on the terminal side

    \(\sin\theta = \frac{y}{r}\), where \(r=\sqrt{x^2+y^2}\).

  • Definition of cosine in terms of a point on the terminal side

    \(\cos\theta = \frac{x}{r}\), where \(r=\sqrt{x^2+y^2}\).

  • Definition of tangent in terms of a point on the terminal side

    \(\tan\theta = \frac{y}{x}\), provided \(x \neq 0\).

  • How to find coterminal angles

    Add or subtract multiples of \(360^\circ\) or \(2\pi\) to the given angle.

  • Use of reciprocal identities to find trig values

    If you know sin, find csc by taking its reciprocal, and similarly for cos/sec and tan/cot.

  • Using Pythagorean identities to find trig values

    Use \(\sin^2\theta + \cos^2\theta = 1\) to find one function if the other is known.

  • Finding angle measures from trigonometric values

    Use inverse trig functions to find angle measures from known trig values.

  • Rules for angle measures in degrees and radians

    360 degrees = \(2\pi\) radians; 180 degrees = \(\pi\) radians.